Student Work from Let's Be Rational, Problem 3.3

Problem 3.3: Sharing a Prize Analysis of Student Work


When given an opportunity to think about how operations work with various numbers in a problem solving setting, students will generate both non-standard and standard ways of calculating. These student strategies (from Let’s Be Rational Problem 3.3) are a teacher’s documentation of student thinking over many years using CMP. Knowing the ways that students may approach the problems can be helpful to teachers in a few ways.

First, a teacher can anticipate and make sense of student reasoning prior to seeing it in the classroom. This can help a CMP teacher prepare possible questions to help students explain and justify mathematical reasoning.

Second, CMP Teachers have found that it is important to help students document their mathematical thinking. This is especially true for students who are new to a problem solving, inquiry-based learning environment.

Third, this is an example of how the classroom strategies may be summarized in a lesson.


Many people have learned to divide fractions by learning an algorithm that may or may not make sense them. In CMP, students are asked to make sense of fraction division on their way to developing an algorithm.

Let’s Be Rational Problem 3.3: Sharing Pizza is a chance for students to develop a strategy for dividing a fraction by a whole number.

Two Artifacts are provided:

  • A copy of Problem 3.3 from the student book and the Focus Question from the Teacher Guide
  • A document with analysis of the student strategies from Problem 3.3


This student work can serve as a guide for planning, teaching, assessing, and reflecting on the mathematics of this Problem. The work can be downloaded for teachers to examine during planning time, for use in a collaborative meeting, or for use in professional development activities. Teachers could also choose to share one or more of the strategies with her/his class and ask them to discuss the validity of the strategy and/or whether it might be generalizable.

Questions to Consider

Planning for the Lesson
  • What strategies do you anticipate students using to accomplish the task?
  • How do these pieces of student work reflect the mathematical goals of the problem? Could these students answer the Focus Question?
  • How does the student influence how you might consider teaching the lesson?
Formative Assessment
  • What mathematical strengths do the students have? Weaknesses?
  • What types of questions might you ask to help these students clarify or extend their thinking?
  • If your students produce work like this, what impact will it have on how you would teach the next few problems?
Planning for the Summary
  • Which of the strategies work for any type of division with fractions, whole numbers, or mixed numbers?
  • What opportunities could be provided for students to compare and contrast the strategies?
  • How might you orchestrate the Summarize of the lesson if your students produce work like these samples?
Teacher Reflections
  • Have the students demonstrated sufficient understanding of the Focus Question posed for this problem?
  • How will this set of student work affect your planning for the next lesson?